Defect Motifs for Constant Mean Curvature Surfaces
Halim Kusumaatmaja and David J. Wales
University Chemical Laboratories, University of Cambridge,
arXiv:1303.6786v1 [cond-mat.soft] 27 Mar 2013
Lensfield Road, Cambridge CB2 1EW, U.K.
(Dated: March 28, 2013)
Abstract
The energy landscapes of electrostatically charged particles embedded on constant mean curvature surfaces are analysed for a wide range of system size, curvature, and interaction potentials. The surfaces are
taken to be rigid, and the basin-hopping method is used to locate the putative global minimum structures.
The defect motifs favoured by potential energy agree with experimental observations for colloidal systems:
extended defects (scars and pleats) for weakly positive and negative Gaussian curvatures, and isolated defects for strongly negative Gaussian curvatures. Near the phase boundary between these regimes the two
motifs are in strong competition, as evidenced from the appearance of distinct funnels in the potential energy
landscape. We also report a novel defect motif consisting of pentagon pairs.
1
Introduction - The distribution of electrostatically charged particles on curved surfaces [1] provides a testing ground for global optimisation algorithms [2–7], as well as useful insights into a
number of materials science and biological applications, including the packing of virus capsids
[8, 9], fullerene structures [10, 11], colloidal crystals [12, 13], and proteins on lipid membranes
[14]. A key issue in understanding the resulting structures is the appearance of defects, which
strongly influence the mechanical, electronic and optical properties. Defect structures have even
been exploited for rational design of templated self-assembly; for example, polar defects have
been used to create divalent metal nanoparticles [15].
Frustration is intrinsic to curved surfaces because of the competition between local order and
long-range geometrical constraints, and defects appear to screen the resulting geometrical stresses,
even for the ground state configuration. For example, while a hexagonal lattice is possible for
a flat surface, this arrangement cannot occur on a sphere without introducing isolated five-fold
disclinations (pentagons; with positive topological charges), or composite structures with sevenfold disclinations (heptagons; negative topological charges) [2–7, 12, 16, 17]. Generally, adjacent
combinations of pentagons and heptagons may appear as topologically uncharged or charged lines
of dislocations, corresponding to pleats (e.g. pentagon-heptagon topological dipoles) and scars
(e.g. pentagon-heptagon-pentagon arrangements), respectively.
Isolated heptagons may also exist if the Gaussian curvature of the substrate is strongly negative,
as was shown recently in studies of two-dimensional colloidal crystals on the surface of capillary
bridges [18]. By systematically varying the shape and thus the curvature of the substrate, a sequence of transitions was observed from zero defects to isolated dislocations, pleats, scars, and
isolated disclinations.
In the present contribution we characterise the energy landscape of electrostatically charged
particles embedded on constant mean curvature surfaces, modelling in particular the recent experimental setup by Irvine et al. for colloidal crystals [18, 19]. Employing the basin-hopping
algorithm [20–23], we identify likely global minimum configurations for a wide range of system
sizes and surface curvatures, considering several different interaction potentials. Not only are we
able to reproduce the experimental sequence of defect transitions, but we also identify a new defect motif corresponding to pentagon pairs, which may appear on surfaces with both positive and
negative Gaussian curvatures. Furthermore, we show here for the first time the hierarchical nature of the potential energy landscape for these systems. Using disconnectivity graphs [24], we
demonstrate the appearance of separate funnels in the landscape, corresponding to competing de2
fect motifs favouring dislocations and disclinations. Our analysis also provides information on the
rearrangement mechanisms between defect patterns, as well as insight into likely thermodynamic
signatures for structural transitions.
Methodology - We consider N identical electrostatically charged particles embedded on the
surface of catenoids (zero mean curvature) and unduloids (non-zero mean curvature) [25]. To represent the interactions of these colloids trapped on fluid interfaces [26], we have mainly considered
P
1
−(rij /λ)
, where rij is the Euclidean distance beYukawa potentials of the form: V = N
j>i rij e
tween particle i and j, and λ is the screening length. We choose λ = 0.1 so that the ratio between
the screening length and the length of the capillary bridges is of the same order as that used in
experiments [18, 26].
We have also considered four other potentials: Yukawa with λ = 0.4, Coulomb, LennardJones, and a repulsive Lennard-Jones form. Qualitatively, the defect motifs are the same provided
that the particle density per unit area is sufficiently high, indicating that our defect analysis should
have wide-ranging applicability to curved surfaces. This regime corresponds to N > 200. If the
number of electrostatically charged particles is smaller, they tend to populate the boundaries when
Coulomb or other similarly long-range potentials are used; for the Lennard-Jones potential, we
observe patches of hexagonal lattice separated by large spaces. We therefore focus on 200 < N <
600 in the present work.
Any point on the surface of an unduloid or a catenoid can be parameterised using two
variables u and v [27]. For catenoids, the mapping to Cartesian coordinates takes the following form (x, y, z) = (c cosh (v/c) cos (u), c cosh (v/c) sin (u), v), where 0 ≤ u < 2π,
−zm ≤ v ≤ zm , and c is a free parameter corresponding to the waist of the catenoid in the
z = 0 plane. The corresponding transformation is more elaborate for unduloids (x, y, z) =
(m sin µv + n)1/2 cos (u), (m sin µv + n)1/2 sin (u), aF (µv/2 − π/4, k) + cE (µv/2 − π/4, k) ,
with µ = 2/(a + c), k 2 = (c2 − a2 )/c2 , m = (c2 − a2 )/2, and n = (c2 + a2 )/2. F (φ, k) and
E(φ, k) are elliptic integrals of the first and second kind. We tune the shape of the unduloids by
varying the parameters a, c, and the range of values for v [27]. For a neck shape unduloid [e.g.
Fig. 1(a-c)], v is centred around vc = 3π/2µ, while for a barrel shape unduloid, vc = π/2µ. The
maximum and minimum values of v are chosen so that −zm ≤ z ≤ zm .
To characterise the most favourable geometries we employ basin-hopping global optimisation
[20–23]. In this method, random geometrical perturbations are followed by energy minimisation,
and moves are accepted or rejected based upon the energy differences between local minima. The
3
minimisation procedure transforms the energy landscape of the system into the set of catchment
basins for the local minima, and makes the basin-hopping method very effective for finding lowlying structures. Further details are provided in the supporting information.
We have also constructed databases of connected minima for selected systems, starting from the
low-lying structures found in the basin-hopping runs. We employed doubly-nudged elastic band
transition state searches [28], where images corresponding to local maxima were tightly converged
to transition states using hybrid eigenvector-following [29, 30]. This procedure provides a global
survey of the potential energy landscape, which we then visualise using disconnectivity graphs
[24]. We find that distinct structural arrangements of the particles result in separate funnels in the
landscape. Moreover, the database of connected minima and transition states allows us to predict
energy barriers and rearrangement mechanisms between defect rearrangements.
To visualise the defect structures, we use Voronoi constructions. Thus, pentagons, hexagons,
and heptagons correspond to particles with five, six, and seven neighbours. All the results presented here were obtained using the GMIN, OPTIM, and PATHSAMPLE programs [31], which
are available for use under the GNU public license.
Defect motifs on unduloids - We first analyse the arrangement of electrostatically charged particles embedded on non-zero constant mean curvature surfaces (unduloids). The results for N = 600
identical particles are presented in Fig. 1. The unduloid parameters a and c were varied while keeping the height (2 zm = 1.5) and volume (V = 2.65) of the unduloids constant. Experimentally,
these parameters correspond to capillary bridges for surfaces with different contact angles [32, 33].
The defect motifs and sequence of transitions were found to be consistent as we varied the number
of particles, and specific results for N = 200 are presented in the supporting information.
Due to the finite number of particles edge effects are an intrinsic feature, and we find that
the first two rows of Voronoi cells from the boundary consistently have more defects. As for
the experimental results on colloidal crystals [18] we find a series of defect transitions, including
isolated heptagons (Fig. 1). The Gaussian curvature is not constant on the surface of an unduloid,
and it is most negative at the waist, where isolated heptagons are located. Far from the waist and
near the edge of the unduloid, we observe lines of dislocations.
For less negative total Gaussian curvatures [Figs 1(b-c)], isolated heptagons disappear and
dislocation lines prevail. The length of the dislocations is also found to correlate strongly with the
curvature: the length is shorter when the curvature is less negative. For weakly negative Gaussian
curvatures, the pentagon-heptagon dipole is a common motif, together with a pair of pentagons.
4
(a) [0.50, 885.89]
(b) [0.60, 8.75]
(c) [0.70, 1.57]
Negative Gaussian Curvature
(f)
(d) [0.75, 0.75]
(e) [0.06, 0.85]
Positive Gaussian Curvature
(g)
(h)
(i)
FIG. 1: Defect motifs for non-zero constant mean curvature surfaces. (a-e) The number of electrostatically
charged particles is N = 600, and the curvature of the capillary bridges is varied while keeping the height
and volume of the bridges constant. The corresponding values for the unduloid parameters [a, c] are given in
square brackets. (f-i) Comparisons to experimental results of Irvine et al. for colloidal crystals, reproduced
(will seek permission) from [18]. Here, red particles have seven neighbors, while yellow particles have five
neighbors. In the experiments, the curvature is tuned by stretching/compressing the liquid bridge, and the
sequence of defect transitions observed is identical to our global optimisation results.
The pentagon pair motif is surrounded by seven hexagons, as for an isolated heptagon, and to the
best of our knowledge, this motif has not been reported before. Since isolated pentagons have a
positive topological charge, it is somewhat surprising that double pentagons may exist in a bound
state, and that they can be favourable for surfaces with negative Gaussian curvature.
We observe no defects for cylinders [zero Gaussian curvature, Fig. 1(d)], as expected. Dislocation lines as well as pentagon pairs then reappear for unduloid with a positive total Gaussian
curvature [Fig. 1(e)]. The orientations of the dislocations and pentagon pairs are reversed for surfaces with positive and negative Gaussian curvature. As predicted by continuum elastic theories
[18, 34], for negative curvatures, the 7-5 dislocation axis runs along the meridian, pointing in
the direction of the boundary. For positive Gaussian curvatures, the dislocation axis points to the
centre of the unduloid instead. Similarly, the pentagon pairs point to the boundary for negative
Gaussian curvatures, and to the centre of the unduloids for positive Gaussian curvatures.
Defect motifs on catenoids - We have also analysed the defect structures for a family of
catenoids (minimal surfaces) with varying values of the waist parameter c. We adjust the height
5
(a) c = 0.4
(b) c = 0.5
(c) c = 0.6
(d) c = 0.7
(e) c = 0.8
Increasingly negative total Gaussian curvature
FIG. 2: Defect motifs on catenoid surfaces for N = 600, with variable waist radius c.
parameter zm so that the radius of the catenoid at zm , c cosh (zm /c) = 1. This procedure results in
less negative total Gaussian curvatures with increasing waist radius c, see Fig. 2.
For strongly negative Gaussian curvature (small c in Fig. 2), we always observe isolated heptagons at the waist of the catenoids, independent of N. We illustrate the disconnectivity graph
[35] for N = 600 and c = 0.40 in Fig. 4(a). The potential energy landscape is clearly hierarchical
in character. Interestingly, there are only two dominant defect configurations near the waist with
a high degree of symmetry, which we label as α and β in the inset, each consisting of eight heptagons. The main variation in the structural arrangements of the ions comes from the dislocation
lines near the edges (see the Supporting Information for animations). The energy barrier for interconverting waist configurations α and β is approximately ∆E1 ∼ 0.4 (in reduced units) if the
configuration near the edge of the catenoid is roughly preserved. On the other hand, the barrier
for rearranging the particle distribution near the edges can be much higher, ∆E2 ∼ 2.2 (reduced
units), as indicated in Fig. 4(a). From the pathway we see that the high barrier is due to global
concerted rotation of several layers of ions near the edges of the catenoids.
For weakly negative Gaussian curvatures (large c in Fig. 2), we never find isolated heptagons,
and only lines of dislocations and pentagon pairs exist. It is worth noting that these defect motifs
have the same orientations as for unduloids with negative Gaussian curvature, above. The energy
landscape for this parameter regime is also simpler. In particular, we find that the number of
possible local minima is considerably smaller.
The transition from defect patterns favouring disclinations to dislocations occurs at moderately
negative Gaussian curvatures, c ∼ 0.55, which we determine by constructing a defect phase diagram as a function of the number of electrostatically charged particles, N, and the catenoid waist
radius, c (Fig. 3). Our global optimisation results further suggest that this transition is only weakly
6
Number of particles N
600
550
500
450
400
350
300
250
200
0.4
0.5
0.6
0.7
0.8
Waist radius c
FIG. 3: Defect phase diagram as a function of the catenoid waist radius, c, and number of electrostatically
charged particles, N . Squares denote global minimum structures containing isolated heptagons, while
circles correspond to structures without isolated heptagons.
dependant on the number of particles in this size range. The phase diagram for the repulsive
Lennard-Jones potential is shown in the Supporting Information, where we find similar behaviour,
except that the phase boundary is shifted to larger c. Our results for N < 600 are probably far
from the continuum limit, where Bowick and Yao [34] predict that c will decrease monotonically
with N. Additionally, the phase boundary found in Fig. 3 is neither smooth nor monotonic. We
can explain this observation by analysing the disconnectivity graph for N = 600 and c = 0.575.
As shown in Fig. 4(b), there are two distinct funnels, one favouring dislocations and the other
favouring disclinations. The energy difference between the lowest configurations in the two funnels is very small compared to the energy barrier (∆E3 ∼ 0.7 in reduced units) associated with
interconverting the two defect motifs.
Discussion - We have investigated the most favourable defect motifs for electrostatically
charged particles embedded on zero and non-zero constant mean curvature surfaces. By varying the curvature of the embedding surface, we have characterised the detailed structures and
energetics of a wide range of defect motifs, including dislocation lines and isolated disclinations,
in excellent agreement with experiment and predictions from continuum elastic theories. The
appearance of a new defect motif consisting of pentagon pairs is also predicted.
Taking the typical experimental values for electrostatic interactions between colloids embedded
on interfaces, the energy scale for one simulation unit corresponds to between 10 and 100kB T (at
room temperature). The energy barriers of interest, as shown in Fig. 4, are therefore quite large
and the structures observed in experiments could be trapped in local minima. For experimentally relevant temperatures the favoured defect motifs are primarily determined by the potential
7
∆E2
∆E3
∆E1
α
α
β
β β
α
α
(a)
(b)
β
disclinations
dislocations
α
FIG. 4: Disconnectivity graphs for N = 600 electrostatically charged particles embedded on catenoids with
waist radius (a) c = 0.40 and (b) c = 0.575. The insets in (a) and (b) show representative minima for the
corresponding funnels in the potential energy landscape. We also indicate the typical energy barriers for
interconverting different defect motifs. ∆E1 , ∆E2 , and ∆E3 are respectively 0.4, 2.2, and 0.7 in reduced
units.
energy, which is consistent with the successful structure predictions that we have reported. We
hope these results will stimulate future theoretical and experimental work. In particular, it will be
interesting to modulate (reduce) the strength of interactions between the electrostatically charged
colloids, so that the defect rearrangement mechanisms and thermodynamics of the system may
be fully explored. Since the energy landscape is hierarchical, with different energy scales for
interconverting the defect motifs, we expect the thermodynamics of the system to be very rich.
For example, we predict that there will be multiple signatures in the heat capacity corresponding
to alternative defect morphologies [23]. Calculations of the thermodynamics, kinetic rates, and
rearrangement pathways for these defect motif transitions are currently underway and will be presented elsewhere. The interplay between the arrangement of the electrostatically charged particles
and possible deformation of the interface is still an open question, and allowing for flexible rather
than rigid curved surfaces is another avenue for future research.
Analog models of the defect motifs presented here can be constructed using Polydron tiles [36],
as illustrated in the Supporting Information (SI). The SI also contains animations of the defect
8
morphologies shown in Figs 1, 2 and 4, the tabulated numbers of positive and negative topological
charges, and detailed analysis of the net topological charges as a function of the integrated Gaussian curvature for several representative cases. These putative global minima found here will be
made available online from the Cambridge Cluster Database [37].
Acknowledgements - This research was supported by EPSRC Programme grant EP/I001352/1
and by the European Research Council.
[1] J. J. Thomson, Philos. Mag. 7, 237 (1904).
[2] L. T. Wille, Nature 324, 46 (1986).
[3] E. L. Altschuler, T. J. Williams, E. R. Ratner, R. Tripton, R. Stong, F. Dowla, and F. Wooten, Phys.
Rev. Lett. 72, 2671 (1994).
[4] T. Erber and G. M. Hockney, Phys. Rev. Lett. 74, 1482 (1995).
[5] T. Erber and G. M. Hockney, Adv. Chem. Phys. 98, 495 (1997).
[6] D. J. Wales and S. Ulker, Phys. Rev. B 74, 212101 (2006).
[7] D. J. Wales, H. McKay, and E. L. Altschuler, Phys. Rev. B 79, 224115 (2009).
[8] D. L. D. Caspar and A. Klug, Cold Spring Harbour, Symp. Quant. Biol. 27, 1 (1962).
[9] C. J. Marzec and L. A. Day, Biophys. J. 65, 2559 (1993).
[10] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, and R. E. Smalley, Nature 318, 162 (1985).
[11] A. Pérez-Garrido, Phys. Rev. B 62, 6979 (2000).
[12] A. R. Bausch, M. J. Bowick, A. Cacciuto, A. D. Dinsmore, M. F. Hsu, D. R. Nelson, M. G. Nikolaides,
A. Travesset, and D. A. Weitz, Science 299, 1716 (2003).
[13] P. Lipowsky, M. J. Bowick, J. H. Meinke, D. R. Nelson, and A. R. Bausch, Nature Mater. 4, 407
(2005).
[14] R. N. Frese, J. C. Pàmies, J. D. Olsen, S. Bahatyrova, C. D. van der Weij-de Wit, T. J. Aartsma,
C. Otto, C. N. Hunter, D. Frenkel, and R. van Grondelley, Biophys. J. 94, 640 (2008).
[15] G. A. DeVries, M. Brunnbauer, Y. Hu, A. M. Jackson, B. Long, B. T. Neltner, O. Uzun, B. H. Wunsch,
and F. Stellacci, Science 315, 358 (2007).
[16] M. Bowick, A. Cacciuto, D. R. Nelson, and A. Travesset, Phys. Rev. Lett. 89, 185502 (2002).
[17] M. J. Bowick and L. Giomi, Adv. Phys. 58, 449 (2009).
[18] W. T. M. Irvine, V. Vitelli, and P. M. Chaikin, Nature 468, 947 (2010).
9
[19] W. T. M. Irvine, M. J. Bowick, and P. M. Chaikin, Nature Mater. 11, 948 (2012).
[20] Z. Li and H. A. Scheraga, Proc. Natl. Acad. Sci. USA 84, 6611 (1987).
[21] D. J. Wales and J. P. K. Doye, J. Phys. Chem. A 101, 5111 (1997).
[22] D. J. Wales and H. A. Scheraga, Science 285, 1368 (1999).
[23] D. J. Wales, Energy Landscapes (Cambridge University Press, Cambridge, 2003).
[24] O. M. Becker and M. Karplus, J. Chem. Phys. 118, 3891 (2003).
[25] C. Delauney, J. Math. Pures Appl., Sér. 1 6, 309 (1841).
[26] M. E. Leunissen, A. van Blaaderen, A. D. Hollingsworth, M. T. Sullivan, and P. M. Chaikin, Proc.
Natl. Acad. Sci. USA 104, 2585 (2007).
[27] M. Hadzhilazova, I. M. Mladenov, and J. Oprea, Archivum Mathematicum 43, 417 (2007).
[28] S. A. Trygubenko and D. J. Wales, J. Chem. Phys. 120, 2082 (2004).
[29] G. Henkelman, B. P. Uberuaga, and H. Jónsson, J. Chem. Phys. 113, 9901 (2000).
[30] L. J. Munro and D. J. Wales, Phys. Rev. B 59, 3969 (1999).
[31] http://www-wales.ch.cam.ac.uk/software.html.
[32] H. Kusumaatmaja and R. Lipowsky, Langmuir 26, 18734 (2010).
[33] E. J. D. Souza, M. Brinkmann, C. Mohrdieck, A. Crosby, and E. Arzt, Langmuir 24, 10161 (2008).
[34] M. J. Bowick and Z. Yao, Europhys. Lett. 93, 36001 (2011).
[35] In the disconnectivity graph, the vertical axis corresponds to the energy of the system. Each line
ends with a local minimum. A node joins minima which can be interconverted without exceeding the
energy of the node (thus providing information on the energy barriers between minima). For clarity,
we discretise the energy in regular steps along the vertical axis.
[36] http://www.polydron.com/.
[37] http://www-wales.ch.cam.ac.uk/˜wales/CCD/ThomsonCMC/table.html.
10